Final answer:
Invariant points, such as the origin in a rotated coordinate system, remain unchanged irrespective of the transformation. These points are crucial in mathematics and physics, serving as stable reference points in various forms of motion and transformations.
Step-by-step explanation:
Invariance Under Coordinate System Rotation
When a coordinate system is rotated, the distances between points in the plane do not change since the rotation does not affect the magnitude of vectors. To prove the invariance of the distance of point P to the origin, we can use the distance formula, which states that the distance from origin to point P in the plane with coordinates (x, y) is given by
. Under rotation to a new system S', the point P will have new coordinates (x', y'), but due to the properties of rotations, the magnitude of the vector from the origin to P remains the same, hence √
, demonstrating that the distance of point P to the origin is invariant under rotations.
Similarly, for points P and Q, the distance between them is calculated using the Pythagorean theorem. If these two points are rotated within the coordinate system, the relative position of P to Q is unaltered. Therefore, the distance between P and Q stays constant, demonstrating invariance under rotation.
Examples of Invariant Points
Origin in a coordinate system rotation
Equilibrium points in physical systems
Reference points in relative motion scenarios
The concept of invariant points is important in both mathematics and physics, as they provide a reference or starting point to which changes can be measured and understood.